Induced orbit data and induced unitary representations for complex groups

We use induced orbit covers to define induced orbit data.By studying the space of regular functions on orbit cover,we know that the induced representation has close connection with the induced orbit datum under the meaning of Vogan’s conjecture.Therefore,when verifying Vogan’s conjecture,many cases can be reduced to the case of rigid orbit data.     

完备稠序线性序拓扑空间上的奇周期轨序关系

研究完备稠序线性序拓扑空间上连续自映射的周期轨,指出当连续自映射有（2n＋1）-周期轨而没有（2n-1）-周期轨时,该（2n＋1）-周期轨上各点的序关系.利用这个关系将Sharkovskii定理从实直线推广到完备稠序线性序拓扑空间上。     

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.     

Martelli’s Chaos in Inverse Limit Dynamical Systems and Hyperspace Dynamical Systems

In this paper we investigate Martelli’s chaos of inverse limit dynamical systems and hyperspace dynamical systems which are both induced from dynamical systems on a compact metric space. We give the implication of Martelli’s chaos among those systems. More precisely, we show that inverse limit dynamical system is Martelli’s chaos if and only if so is original system, and we prove that hyperspace dynamical system is Martelli’s chaos implies original system is Martelli’s chaos if the orbit of every single point set of original system is unstable in hyperspace dynamical system.     

Non-discrete affine buildings and convexity

Affine buildings are in a certain sense analogs of symmetric spaces. It is therefore natural to ask for analogs of results for symmetric spaces. We prove a version of Kostant?s convexity theorem for thick non-discrete affine buildings. Kostant proves that the image of a certain orbit of a point x in the symmetric space under a projection onto a maximal flat is the convex hull of the Weyl group orbit of x. We obtain the same result for a projection onto an apartment in an affine building. The methods are mostly borrowed from metric geometry. Our proof makes no appeal to the automorphism group of the building. However the final result has an interesting application for groups acting nicely on non-discrete buildings, such as groups admitting a root datum with non-discrete valuation. Along the proofs we obtain that segments are contained in apartments and that certain retractions are distance diminishing.     

On the spectral theory of singular indefinite Sturm-Liouville operators

We consider a singular Sturm-Liouville differential expression with an indefinite weight function and we show that the corresponding self-adjoint differential operator in a Krein space locally has the same spectral properties as a definitizable operator.     

Isoparametric submanifolds in two-dimensional complex space forms

We show that an isoparametric submanifold of a complex hyperbolic plane, according to the definition of Heintze, Liu and Olmos’, is an open part of a principal orbit of a polar action. We also show that there exists a non-isoparametric submanifold of the complex hyperbolic plane that is isoparametric according to the definition of Terng’s. Finally, we classify Terng-isoparametric submanifolds of two-dimensional complex space forms.     

Free continuous actions on zero-dimensional spaces

We show that every countably infinite group admits a free, continuous action on the Cantor set having an invariant probability measure. We also show that every countably infinite group admits a free, continuous action on a non-homogeneous compact metric space and the action is minimal (that is to say, every orbit is dense). In answer to a question posed by Giordano, Putnam and Skau, we establish that there is a continuous, minimal action of a countably infinite group on the Cantor set such that no free continuous action of any group gives rise to the same equivalence relation.     

纤维拓扑空间理论简介

Fibrewise topological spaces theory,presented in the recent 20 years,is a new branch of mathematics developed on the basis of General Topology,Algebra topology and Fibrewise spaces theory.It is associated with differential geometry,Lie groups and dynamical systems theory.From the perspective of Category theory,it is in the higher category of general topological space,so the discussion of new properties and characteristics of the variety of fibre topological space has more important significance.This paper introduces the process of the origin and development of Fibrewise topological spaces theory.Then,we study the main contents and important results in this branch.Finally,we review the research status of Fibrewise topological spaces theory and some important topics.     

向量场的旋度判定平面(空间)微分系统闭轨的不存在性

利用向量场的旋度 ,我们可判定平面、空间微分系统闭轨的不存在性 ,得到类似 Bendixson,Dulac判断法的一种新判定法 .同时还得到平面微分系统闭轨存在时的相对位置     

Poisson transforms adapted to BGG-complexes

We present a new construction of Poisson transforms between vector bundle valued differential forms on homogeneous parabolic geometries and vector bundle valued differential forms on the corresponding Riemannian symmetric space, which can be described in terms of finite dimensional representations of reductive Lie groups. In particular, we use these operators to relate the BGG-sequences on the domain to twisted deRham sequences on the target space. Finally, we explicitly design a family of Poisson transforms between standard tractor valued differential forms for the real hyperbolic space and its boundary which are compatible with the BGG-complex.     

一类太空等离子体单粒子运动模型的同宿轨

由于空间等离子体间接观察的不确定性和不可重复性以及只能进行被动实验的局限性,为了准确地描述空间等离子体粒子运动的动力学特征,建立了一类太空等离子体单粒子运动的非线性模型．首先运用重合度理论探讨了一类非线性问题的周期解,然后将其应用于太空等离子体单粒子运动模型的同宿轨问题的研究,获得了该模型存在同宿轨的新结果,为研究空间环境提供了更好的观测和理论基础.     

Infinite positive-definite quadratic programming in a Hilbert space

This note generalizes the results of Benson, Smith, Schochetman, and Bean (Ref. 1) regarding the minimization of a positive-definite functional over the countable intersection of closed convex sets in a Hilbert space. A finite approximating subproblem for the general case is shown to have the same strong convergence properties of the earlier work without any of the specialized structures imposed therein. In particular, the current development does not rely on any properties ofl ² and does not require the Hilbert space to be separable.     

Linear Optimization with Box Constraints in Banach Spaces

Let X be a partially ordered real Banach space, let a,b∈X with a≤b. Let φ be a bounded linear functional on X. We say that X satisfies the box-optimization property (or X is a BOP space) if the box-constrained linear program: max 〈φ,x〉, s.t. a≤x≤b, has an optimal solution for any φ,a and b. Such problems arise naturally in solving a class of problems known as interval linear programs. BOP spaces were introduced (in a different language) and systematically studied in the first author’s doctoral thesis. In this paper, we identify new classes of Banach spaces that are BOP spaces. We present also sufficient conditions under which answers are in the affirmative for the following questions:

1. (i)
   When is a closed subspace of a BOP space a BOP space?
    
2. (ii)
   When is the range of a bounded linear map a BOP space?
    
3. (iii)
   Is the quotient space of a BOP space a BOP space?
    

     

On the tangent flow of a stochastic differential equation with fast drift

We investigate the behavior of the tangent flow of a stochastic differential equation with a fast drift. The state space of the stochastic differential equation is the two-dimensional cylinder. The fast drift has closed orbits, and we assume that the orbit times vary nontrivially with the axial coordinate. Under a nondegeneracy assumption, we find the rate of growth of the tangent flow. The calculations involve a transformation introduced by Pinsky and Wihstutz.

     

Harmonic forms and cohomology of singular stratified spaces

We construct a potential theory for differential forms on compact stratified spaces, and we show that the space of harmonic forms of degree k is isomorphic to the singular cohomology of degree k. This theory applies to any triangulable topological space.     

On the hilbert boundary value problem for holomorphic function in sobolev spaces

The holomorphic solution φ of the Hilbert boundary value problem Re (a + ib) φ = g on γ in a disk D bounded by the simple closed curve γ has been solved in the space of Hölder-continuously differentiable functions C₁, C_1,α (D) by many authors. Under the assumptionthat g belongs to the Slobodecky space W_s,p (γ), s = 1 ? 1/p,1 < p < ∞ it is shown here that the problem has a uniquesolution in the Sobolev space W_1,p: (D). An a-priori estimate for the norm of in W_{1 p}(D)is given.     

Connected 4-dimensional stable planes with many central collineations

This paper gives a complete classification of all connected 4-dimensional stable planes with the property that every point is the centre of a non-trivial central collineation. It is shown that under these assumptions the automorphism group has an open orbit on the point space. This implies the existence of an open subplane that carries the additional structure of a generalized symmetric space in the sense of differential geometry. Now the classification of all 4-dimensional generalized symmetric planes yields the desired classification.Dedicated to Professor H. Salzmann on his 60th birthday     

Metric,Banach, and Hilbert Spaces of <Emphasis Type="Italic">φB</Emphasis>-Distributions

We introduce φ-distributions and prove that their set is ametric space. We also consider a Banach space and a Hilbert space of such distributions. The results are applied to differential equations with Laurent’s type coefficients.